Mathematical modelling identifies conditions for maintaining and escaping feedback control in the intestinal epithelium

The intestinal epithelium is one of the fastest renewing tissues in mammals. It shows a hierarchical organisation, where intestinal stem cells at the base of crypts give rise to rapidly dividing transit amplifying cells that in turn renew the pool of short-lived differentiated cells. Upon injury and stem-cell loss, cells can also de-differentiate. Tissue homeostasis requires a tightly regulated balance of differentiation and stem cell proliferation, and failure can lead to tissue extinction or to unbounded growth and cancerous lesions. Here, we present a two-compartment mathematical model of intestinal epithelium population dynamics that includes a known feedback inhibition of stem cell differentiation by differentiated cells. The model shows that feedback regulation stabilises the number of differentiated cells as these become invariant to changes in their apoptosis rate. Stability of the system is largely independent of feedback strength and shape, but specific thresholds exist which if bypassed cause unbounded growth. When dedifferentiation is added to the model, we find that the system can recover faster after certain external perturbations. However, dedifferentiation makes the system more prone to losing homeostasis. Taken together, our mathematical model shows how a feedback-controlled hierarchical tissue can maintain homeostasis and can be robust to many external perturbations.

Next, we calculate the defects χ 2 of model 2 for the three perturbations. For the case of the first perturbation, the defect can be directly obtained by solving the system analytically for S(0) =S, D(0) = 0 and calculating the integral. The other two defects are obtained by using the first-order approximation of the dynamics derived earlier and analytically calculating their integrals after choosing the respective initial conditions. We get: For model 3, we again only derive the relaxation dynamics after the three perturbations, and will examine them numerically. They are given by: where d := ω/2, and r := ω(−4β 0 + 4δ + ω)/2.
For model 4, we can analytically calculate the defects after the three perturbations. They are: Third perturbation.  Columns represent three different scenarios with different steady-state stem cell fractions of 1%, 10%, and 25%, respectively.

IV. DEDIFFERENTIATION CAN LIMIT OSCILLATORY BEHAVIOUR
The modified model permits exactly one non-trivial steady-state at and its Jacobian at this steady-state is given by This matrix has eigenvalues If oscillations occur, we have complex eigenvalues, which we can split into a real and an imaginary part as follows: For the colon epithelium model without dedifferentiation the real part of the complex eigenvalues of the Jacobian at steady-state was given by −δ 0 ω/(2β). Accordingly, additionally allowing for dedifferentiation reduces the real part by β 0 /(2ω) > 0, hence always causing a faster decay of the oscillations after perturbations.
For the model without dedifferentiation we had an imaginary part of the eigenvalues of 4β 3 ω − 4β 2 δ 0 ω − δ 2 0 ω 2 /(2β). Hence, adding the dedifferentiation to the model changes the radicand of the imaginary part of the eigenvalues by Thus, we can find a critical value * 0 , which, when exceeded by 0 will reduce the frequency of oscillations after perturbations. It is given by 10 In other words, adding a linear dedifferentiation function will speed up the amplitude decay of the oscillations after perturbations and can also, in case of a sufficiently big basal dedifferentiation rate, decrease the frequency of these oscillations.
We can generalise this finding to arbitrary decreasing differentiable functions . To this end, we construct a Taylor expansion of around the steady-state of the form (D) ≈ a + bD + O(D 2 ) with a, b ∈ R. This way, in a sufficiently small neighbourhood around the steady-state, the system behaves as if was linear; and because by definition is always positive and monotonically decreasing, clearly b < 0 and accordingly a > 0. Hence, the argument for linear functions we made previously also applies here when we simply replace 0 with a.

V. CONVERGENCE OF THE COLON EPITHELIUM MODEL WITH DEDIFFERENTIATION TO A STABLE CELL TYPE RATIO
For sufficiently high population sizes, the dynamics of the system are governed by the set of linear differential equations dS(t) dt = βS(t) − δ max S(t) + min D(t) dD(t) dt = δ max S(t) − min D(t) − ωD(t).